[Chaste]

Considering the phosphine group is at the axial position, it will compete for pi back donation with the CO trans to it. Whereas, it may be highly symmetrical, thus I am guessing 2 CO stretches. but a suggested answer quotes 3. Care enlightening me?

[cheese (MSW)]

There is only one form of Cr(CO)5(PPh3) so saying P is axial is not correct.
You need to use group theory to first put Cr(CO)5P in its correct point group,
and then determine the number of ν(CO).  Two ν(CO) is not the correct answer.
PPh3 is a weak π-acceptor ligand compared to CO so how will the CO stretches
shift: Cr(CO)6→Cr(CO)5P?

[cheese (MSW)]

I sketched out a whole lecture on the interpretation of the IR spectra of metal carbonyl cmpds but realized it wouldn’t be appreciated.  Suffice it to say that the rigorous determination of the number of ν(CO) bands a metal-CO cmpd will exhibit is determined by group theory. This subject is usually covered as an elective in 3rd/4th y undergrad chem., or even grad level (it is not trivial suibject like the C-13 NMR spectra of M-CO cmpds).  For C4v Cr(CO)5PPh3 three bands are predicted from group theory (the PPh3 ligand is considered as a sphere).   Couldn’t find data for Cr(CO)5PPh3, but for Mo(CO)5PPh3 the CO stretches appear at 2073, 1984, 1952 cm^-1 in agreement with theory.
When comparing the σ-donor/ π-acceptor properties of PR3 ligand use the lowest energy (E) mode and compare it to the single ν(CO) at ~2000 cm^-1 in M(CO)6 (M = Cr, Mo, W).

[Chaste]

Is it possible to deduce it with logic without the use of group theory?

Since p is not at its axial position but the equatorial, the CO group trans to it will be affected the greatest. That would constitute to one stretch. Does that also put the other COs at the equatorial and axial position inequivalent? Giving in total 3 stretches?

[cheese (MSW)]

Your so-called "axial" and "equatorial"  Cr(CO)5P are the same molecule.  There is only one geometry for
M(CO)5L  (making a model helps).  It is a common mistake.

[Lambin]

I have a two questions regarding the application of group theory in predicting the IR absorption bands of v(CO).

1. Where does the C2 rotational axis exist in the C4v point group of Cr(CO)5PPH3?
2. In forming the reducible representation for v(CO), I've obtained 5, 1, 0, 1, 3 for the symmetry operations respectively. However, N was enumerated as a non-integral value when I was attempted to determine the number of times A1 occurs for v(CO). Where did I make an error?

[cheese (MSW)]

The C2 axis is the same axis as the C4 (P-Cr-CO).  Your reducible representation is thus 5, 1, 1, 1, 3:
which reduces to 2A1, B2(or B1) and E.   I am very rusty with my group theory but this should work. 

[Lambin]

If the 2C4 principal rotation axes represent the clockwise and counterclockwise C4 rotation about P-Cr-CO, then why isn't the C2 rotation represented in the same manner? In other words, why is there only one rotational axis about the C2?

[cheese (MSW)]

It has been a number of years since I studied the details of Group Theory but let me take, a hopefully intelligent, stab at the last question.  The matrix representation for the symmetry operation of a rotation clockwise about a C4 axis (+90°, π/2) is distinct from the anticlockwise C4 rotation (by -90°, -π/2) and therefore clockwise and anticlockwise rotation must be considered as different symmetry elements.  On the other hand, the matrix representation for the C2 symmetry operation ±180°, ±π is identical whether carried out clockwise or anticlockwise and therefore unique.